Prove that the mean of any four consecutive even integers is an integer.
- OCR - GCSE Maths - Question 13 - 2019 - Paper 6
Question 13
Prove that the mean of any four consecutive even integers is an integer.
Worked Solution & Example Answer:Prove that the mean of any four consecutive even integers is an integer.
- OCR - GCSE Maths - Question 13 - 2019 - Paper 6
Step 1
Let the four consecutive even integers be: 2n, 2n + 2, 2n + 4, and 2n + 6
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Answer
To prove that the mean is an integer, we start by expressing the four consecutive even integers as:
First integer: 2n
Second integer: 2n+2
Third integer: 2n+4
Fourth integer: 2n+6
where n is any integer.
Step 2
Calculate the mean of the four integers
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Answer
The mean of these four integers is calculated as follows:
ext{Mean} = rac{(2n) + (2n + 2) + (2n + 4) + (2n + 6)}{4}
Combining the terms in the numerator gives:
= rac{8n + 12}{4} =2n+3
Step 3
Show that the mean is an integer
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Answer
Since n is an integer, multiplying n by 2 gives 2n, which is also an integer. Adding 3 to any integer yields:
2n+3
This is an integer, thus proving that the mean of any four consecutive even integers is an integer.
